arXiv:physics/0302099v1 [physics.geo-ph] 28 Feb 2003
GEOPHYSICAL RESEARCH LETTERS, VOL. , NO. , PAGES 1–4,
Nonlinear phenomena in fluids with temperature-dependent
viscosity: an hysteresis model for magma flow in conduits
Antonio Costa1
Dipartimento di Scienze della Terra e Geologico-Ambientali, Università di Bologna, Italy
Giovanni Macedonio
Osservatorio Vesuviano, Istituto Nazionale di Geofisica e Vulcanologia, Napoli, Italy
Abstract. Magma viscosity is strongly temperature-dependent.
When hot magma flows in a conduit, heat is lost through the walls
and the temperature decreases along the flow causing a viscosity
increase. For particular values of the controlling parameters the
steady-flow regime in a conduit shows two stable solutions belonging either to the slow or to the fast branch. As a consequence, this
system may show an hysteresis effect, and the transition between
the two branches can occur quickly when certain critical points are
reached. In this paper we describe a model to study the relation between the pressure at the inlet and the volumetric magma flow rate in
a conduit. We apply this model to explain an hysteric jump observed
during the dome growth at Soufrière Hills volcano (Montserrat), and
described by Melnik and Sparks [1999] using a different model.
1. Introduction
Like other liquids of practical interest (e.g. polymers, oils),
magma has a strong temperature-dependent viscosity. It is well
known that the flow in conduits of this kind of fluids admits one or
more solutions for different ranges of the controlling parameters.
The problem of the multiple solutions and their stability was studied by Pearson et al. [1973] and by Aleksanopol’skii and Naidenov
[1979]. More recently this phenomenon, applied to magma flows,
was investigated with more sophisticated models by Helfrich [1995]
and by Wylie and Lister [1995]. In Skul’skiy et al. [1999] a simple
method to find the conditions of multiple solutions was adopted and
an experimental study to verify the hysteresis phenomenon predicted
for this process was performed.
2. The model
In this paper we investigate a simple one-dimensional flow model
of a fluid with temperature-dependent viscosity, with the essential
physical properties characterizing the phenomenon of the multiple
solutions and hysteresis as in Skul’skiy et al. [1999]. The fluid flows
in a conduit with constant cross section and constant temperature at
the wall boundaries. We assume that the fluid properties are constant
with the temperature except for the viscosity, and we neglect the heat
conduction along the streamlines, and the viscous heat generation.
Moreover, we assume a linear relation between the shear stress and
1 Also at Dipartimento di Scienze della Terra, Università degli Studi di
Pisa, Italy.
Copyright 2002 by the American Geophysical Union.
Paper number 2001GL014493.
0094-8276/14/2001GL014493$5.00
the strain rate (Newtonian rheology). This last assumption is introduced to simplify the model and allows us to demonstrate that the
multiple solutions in conduit flows are the direct consequence of
the viscosity increase along the conduit induced by cooling (under
particular boundary conditions). Under these hypotheses, the equations for momentum and energy balance for the one-dimensional
steady flow in a long circular conduit (R ≪ L) at low Reynolds
number are:
1 ∂
∂v
µ(T )r
r ∂r
∂r
=
dP
dz
(1)
∂T
∂T
1 ∂
kr
= ρcp v
(2)
r ∂r
∂r
∂z
where R is the conduit radius, L conduit length, r radial direction,
z direction along the flow, v velocity along the flow (we assume
v = v(r, z) ≈ v(r)), cp specific heat at constant pressure, k thermal conductivity, ρ density, µ viscosity, and P pressure, and T
temperature. For magma, the dependence of the viscosity on the
temperature is well described by the Arrhenius law:
µ = µA exp(B/T )
(3)
where µA is a constant and B the activation energy. In this paper,
for simplicity, we approximate eq. (3) by the Nahme’s exponential
law, valid when (T − TR )/TR ≪ 1, where TR is a reference temperature:
µ = µR exp [−β(T − TR )]
(4)
with β = B/TR2 and µR = µA exp(B/TR ). Following Skul’skiy
et al. [1999], we introduce two new variables: the volumetric flow
rate
Q = 2π
Z
R
v(r)rdr
(5)
0
and the convected mean temperature:
2π
T (z) =
Q
∗
Z
R
T (r, z)v(r)rdr
(6)
0
To satisfy the mass conservation, the volumetric flow rate Q is constant along the flow. Integrating eq. (1) and (2) and expressing the
solutions in terms of Q and T ∗ , we obtain:
πR4 exp [β(T ∗ − Tw )] dP =Q
8µw
dz
(7)
∂T ∗
= 2πRα(Tw − T ∗ )
(8)
∂z
where Tw is the wall temperature (taken here as reference, Tw =
TR ), µw fluid viscosity at T = Tw , and α the coefficient of heat
ρcp Q
2
COSTA AND MACEDONIO : HYSTERESIS EFFECTS IN MAGMA FLOWS
transfer through the walls defined as:
α=
k
∂T Tw − T ∗ ∂r r=R
(9)
In the averaged model (eq. 7), we have adopted
µ ≈ µR exp [−β(T ∗ − TR )]
(10)
and as pointed out in Helfrich [1995], the value of β in eq. (10), is
usually smaller than the actual value of β in the fluid (eq. 4).
Eq. (7) and (8) with the boundary conditions:
P (0) = P0 ,
P (L) = 0,
T ∗ (0) = T0
(11)
give an approximate solution of the problem. These equations are
similar to the model of Pearson et al. [1973] (for a plane flow) and
were used by Shah and Pearson [1974] to study the viscous heating
effects. In the nondimensional form, eq. (7) and (8) are:
dp
= −qe−θ
dζ
q
(12)
dθ
+θ =0
dζ
Figure 2. Plot of the experimental results (crosses) of Skul’skiy et al.
[1999]. The dashed line follows the “history” of the flow regimes
imposed to the fluid; the full line represents the path predicted by
the model with the same parameters of the experiment. Modified
after Skul’skiy et al. [1999].
p(ζ) − p0 = q
where
R0
ζ
exp(−Be−ζ/q )dζ
(15)
θ(ζ) = B exp(−ζ/q)
ρcp Q
,
2πRLα
ρcp R3
p=
P,
16µw L2 α
q=
ζ=
z
L
and, therefore, the relation between the nondimensional pressure
(13)
at the conduit inlet p0 and the nondimensional flow rate q is:
θ = β(T ∗ − Tw )
p0 = q
The boundary conditions (11) are rewritten for the new variables:
p(0) = p0 =
ρcp R3 P0
,
16µw L2 α
p(1) = 0,
θ(0) = B
(14)
with B = β(T0 − Tw ). The solutions of eq. (12), satisfying eq. (14)
at the boundaries, are:
Z
1
exp(−Be−ζ/q )dζ
(16)
0
In Fig. 1 we plot relation (16) obtained numerically. We observe
that for values of B greater than a critical value Bc ≃ 3, there are
values of p0 which correspond to three different values of q. By using a simpler model, Skul’skiy et al. [1999] found Bc = 4, whereas
Helfrich [1995] found Bc = 3.03, in good agreement with Pearson et al. [1973]. Moreover, Shah and Pearson [1974] showed that
when the viscous heat generation is important, the value of Bc can
be lower, but for high values of B, the relation between p0 and q is
similar to the case without viscous heat generation.
The stability analysis of the three branches (slow, intermediate
and fast) shows that the intermediate branch is never stable. Moreover, one part of the slow branch is stable to two-dimensional perturbations but unstable to the three-dimensional ones, in a way similar
to the Saffman-Taylor instability [Wylie and Lister, 1995]. In the
case of multiple solutions, an hysteresis phenomenon may occur, as
proposed in Wylie and Lister [1995] and verified experimentally in
Skul’skiy et al. [1999].
In the experiments of Skul’skiy et al. [1999] a fluid with prescribed temperature and pressure was injected into a capillary tube
Table 1. Typical Values of the Parameters Used in This Paper
Parameter
Figure 1. Plot of the relation between the nondimensional flow rate
q and the nondimensional pressure at conduit inlet p, for different
Rock density
Magma density
Conduit radius
Conduit length
Magma temperature
Magma specific heat
Magma thermal conductivity
Symbol
Value
Unit
ρr
ρm
R
L
T0
cp
k
2600
2300
15
5000
1123
1000
2
Kg m−3
Kg m−3
m
m
K
J Kg−1 K−1
W m−1 K−1
COSTA AND MACEDONIO : HYSTERESIS EFFECTS IN MAGMA FLOWS
with constant wall temperature and controlled fluid pressure and
flow rate. The device is used to show the transition between the two
regimes corresponding to the upper and the lower branches.
A comparison between the experimental results (crosses) and theory (full line) is shown in Fig. 2 for the nondimensional variables p0
and q, for B = 4.6. The dashed lines indicate the pressure history
prescribed in the experiments. The two steady-state regimes corresponding to the slow and to the fast branch were clearly recorded.
Starting with a low pressure configuration (point A) and by increasing the pressure, the flow rate increases along the slow branch until
it reaches a critical point (point B). Here, a jump to the fast branch
occurs (point C). Increasing the pressure further, the flow rate increases along this branch, whereas, by decreasing the pressure, the
flow rate decreases moving along the upper branch, until it reaches
another critical point (point D) where the jump on the slow branch
occurs (point E). On the slow branch the nondimensional flow rate is
more than one order of magnitude lower than that on the fast branch.
3. Application
During some basaltic fissure eruptions in Hawaii and in Iceland,
the eruption begins with a rapid opening at high flow rate and, after
few hours, the flow rate quickly decreases. To explain this phenomenon, Wylie and Lister [1995] and Helfrich [1995] proposed
a model similar to the model presented in this paper, based on the
hysteric jump between the fast branch and the slow branch when
the pressure driving the eruption decreases.
A similar phenomenon, showing a jump in the mass flow rate,
was observed during the dome growth (1995-1999) at Soufrière Hills
volcano in Montserrat as described by Melnik and Sparks [1999].
The model used by Melnik and Sparks [1999] to explain this effect is essentially based on the crystal growth kinetics which affects
magma viscosity, and the mechanical coupling between the gas and
the melt through the Darcy law.
In this paper we explain the same phenomenon in terms of the
viscosity variation governed by cooling along the flow. However,
since the crystal content is physically related to the magma temperature, the two models are physically related.
Using the data of Tab. 1, we fit the curve of eq. (16) with the
observed values of the discharge rates and dome height reported in
Melnik and Sparks [1999].
The variation of the dome height reflects a change of the exit
pressure and, as a consequence, a variation of the difference between the inlet and the outlet pressures. Since we assume a zero
exit pressure in our model and the gravity term was not explicitly
accounted for in eq. (1), the variable p0 represents the overpressure
at the base of the conduit (to obtain the actual value, the hydrostatic
pressure ρg(L + H) must be added to its value, where L is the
conduit length and H is the dome height).
The values of α, B and µw are chosen by least square best fitting of the observed data, and the wall temperature Tw was defined
as the temperature for which magma ceases to flow. Fig. 3 shows
the results of least square fitting: the discharge rate is reported on
the x-axis and the overpressure on the y-axis. The crosses indicate
the observed values, reported in Melnik and Sparks [1999], and the
dome height expressed in terms of overpressure.
The values obtained by the best fit of the observed data are:
αbest = 3.56
W m−2 K−1
µbest
= 5 × 108 Pa s
w
B
best
(17)
= 3.46
Fig. 3 shows the agreement of the model with the observed data; the
proposed model is able to explain the hysteresis effect observed in
dome growth at Soufriere Hill (Montserrat) by Melnik and Sparks
[1999], and modeled in a different way.
3
A typical value of the viscosity is µ0 = µ(T0 ) = 107 Pa s for
Montserrat andesite at 1123 K with 4% water content [Melnik and
Sparks, 1999]. This value is in good agreement with eq. (17);
in fact, for B = Bbest and assuming Tw ≈ 873 K, we have
µ0 = µbest
exp (B) ≃ 1.6 × 107 Pa s. Moreover, for B = Bbest
w
and T0 − Tw ≈ 250 K gives β ≃ 0.014 K−1 (for example, from
data of Hess and Dingwell [1996] for a magma with a similar composition and 4% water content, we obtain β ≃ 0.016 K−1 ). Finally,
for the heat transfer coefficient we have:
α≈
k
≈ 4 W m−2 K−1
δT
(18)
where δT is the thermal boundary layer of the flow, while using
k = 2 W m−1 K−1 , δT ≈ 50 cm (Bruce and Huppert [1989] used
δT ≈ 10 cm for dyke length of about one kilometer).
Moreover, we verify the basic assumptions of the model: the
assumption of one-dimensional flow is based on the small diameter/length ratio of the conduit (R/L ∼ 10−3 ), and the small Reynold
Figure 3. Relation between the discharge rate and the pressure. Full
line represents the model prediction; crosses represents observed
values at Soufrière Hills from Melnik and Sparks [1999].
4
COSTA AND MACEDONIO : HYSTERESIS EFFECTS IN MAGMA FLOWS
number is simply verified:
Re =
ρRv̄
2300 × 15 × 0.003
≈
∼ 10−5
µ0
107
(19)
The viscous heating effects can be neglected because the Nahme
number based on the shear stress is small:
G=
)2 R 4
β(− dP
dx
≈1
4kµw
(20)
The assumption of negligible heat conduction along the streamlines is justified by the high value of the Peclet number (the ratio between the advective and the conductive heat conduction):
Pe = (ρcp vR)/k ≈ 105 .
The existence of the multiple solutions for the steady flow allows the system to show a pulsating behavior between the different
solutions. In the case where the initial pressure, is greater than the
critical pressure corresponding to point E in Fig. 4, the system is on
the fast branch of the solution, such as in point A. If the pressure decreases, the system moves along this branch up to the critical point
C. In C a jump to the slow branch occurs (point D). If the pressure
continues to decrease, the discharge rate tends to zero. Instead, if
the pressure increases, the system moves along the slow branch up
to the other critical point E. In this point the jump occurs on the fast
branch and the system reaches point B.
The overpressure conditions and pulsating activity, typical of
dome eruptions, are evident not only at Soufrière Hills, but also
in Santiaguito (Guatemala), Mount Unzen (Japan), Lascar (Chile),
Galeras (Colombia) and Mount St. Helens (USA) [Melnik and
Sparks, 1999].
4. Conclusion
Magma flow in conduits shows the existence of multiple solutions, like other fluids with strong temperature dependent viscosity.
This is a consequence of the increase of viscosity along the flow
due to cooling. For a given pressure drop along the conduit, one or
two stable regimes (fast and slow branches) may exist. The transition between the two branches occurs when critical values are
reached, and an hysteresis phenomenon is possible. These jumps
were evident during the dome growth in the 1995-1999 Soufrière
Hills (Montserrat) eruption. The pulsating behaviour of the dome
growth was previously modeled by Melnik and Sparks [1999] in
terms of the nonlinear effects of crystallization and gas loss by permeable magma.
In this paper we propose a model to describe the nonlinear jumps
between the two stable solutions as a consequence of the coupling
between the momentum and energy equation induced by the strong
temperature-dependent viscosity of magma.
However, since the crystal content is a consequence of cooling,
the two models, although different, are physically related.
Acknowledgments. This work was supported by the European Commission (Contract ENV4-CT98-0713), with the contribution of the Gruppo
Nazionale per la Vulcanologia INGV and the Department of the Civil Protection in Italy.
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A. Costa, Dipartimento di Scienze della Terra e Geologico-Ambientali,
Università di Bologna, Via Zamboni 67, I-40126 Bologna, Italy. (e-mail:
[email protected])
G. Macedonio, Osservatorio Vesuviano - INGV, Via Diocleziano 328,
I-80124 Napoli, Italy. (e-mail: [email protected])
(Received November 30, 2001.)